\chapter{Pattern forming nonlinear optics} % (fold)
\label{cha:patterns}

Transverse optical patterns are fundamental to the operation of the all-optical switch I present in this thesis. This chapter serves to outline the minimum ingredients needed to observe transverse patterns in a nonlinear optical system. The primary requirement for transverse pattern formation is an mechanism that provides significant gain for components of a wave that propagate off-axis. If such off-axis gain is large enough, the stable state of the waves propagating in the medium can become unstable and exhibit 

What are the minimum ingredients needed to observe transverse optical patterns?

Transverse optical patterns are critical for my switching work

Why counterpropagating beams? threshold is lowest for cprop beams!

experimental motivation, and show my pattern data.

brief review of exp and theory (see previous point also, show my data here.)

overview physical effects that give rise to patterns

get to firth and pare

mention gaeta extension

(firth and pare include cross-wave coupling, as well as F4WM and B4WM. This is the difference)

Add back-of-the envelope calculation of theta based on wave wave retardation.

This system has been shown to exhibit rich spatio-temporal dynamics, including phase conjugate reflectivity, self-oscillation, chaos, and pattern formation [REFS].

A pair of optical beams counterpropagating through a transparent nonlinear medium is a simple experimental system that exhibits rich spatio-temporal dynamics. The all-optical switch presented in this thesis relies on instabilities that arise in this configuration. In this chapter, I present several theoretical treatments of the counterpropagating beam system. This discussion serves two purposes, first to introduce the key concepts involved in describing how optical beams couple via nonlinear interactions, and second to outline the historical background of theoretical treatments for this system.

The first treatment, originally presented by Silberberg and Bar-Joseph \cite{Silberberg_1984aa}, models the coupling between two scalar plane-wave beams in a nonlinear medium, and describes the conditions required for energy transfer between the two beams. This model, typically known as two-beam coupling, assumes a medium with non-instantaneous nonlinear response and exhibits instability in the intensity of the counterpropagating beams. The second treatment I will discuss examines four-wave mixing in counterpropagating systems and describes gain and phase conjugate reflection experienced by a third beam interacting with the medium. This treatment, given by Yariv and Pepper \cite{Yariv_1977aa}, assumes that the counterpropagating fields are scalar plane waves, are undepleted (\emph{i.e.} they are constant throughout the medium). The third model, that of Firth and Par\'e, considers the effects of additional degrees of freedom by allowing the transverse dimensions to exhibit instability. Including transverse dimensions reduces the threshold for instability onset below that originally shown by Yariv and Pepper, and exhibits instabilities even in a medium with instantaneous nonlinear response \cite{Firth_1988aa}. This two-dimensional model is the basis for my numerical simulation of optical switching discussed in Chapter~\ref{cha:nummodel}.

\section{Two-beam coupling} % (fold)
\label{sec:two_beam_coupling}



In Chapter~\ref{cha:opt_switching}, I discussed how the refractive index experienced by one wave can be modified by the intensity of another wave when the two propagate together within a nonlinear medium. Now, in order to explore how this coupling leads to instability in the intensity of each beam, consider the situation illustrated in Fig.~\ref{fig:silberberg_setup} \cite{Boyd_2002aa,Silberberg_1984aa}. Two plane-waves with angular frequencies $\omega_1$ and $\omega_2$, wavevectors $\mathbf{k}_1$ and $\mathbf{k}_2$, and relative propagation angle $\theta$, enter a nonlinear medium. The total optical field within the medium can be described by
\begin{equation}
  \widetilde{E}(\mathbf{r},t)=A_1e^{i(\mathbf{k}_1\cdot\mathbf{r}-\omega_1 t)} + A_2e^{i(\mathbf{k}_2\cdot\mathbf{r}-\omega_2 t)} + \mathrm{c.c.},
\end{equation}
where $k_i=n_0 \omega_i/c$, with $n_0$ denoting the linear refractive index experienced by both waves. The intensity distribution within the medium is hence given by
\begin{equation}
  \label{eqn:intensity_pattern}
  I=\frac{n_0 c}{2\pi}\left[A_1A^*_1+A_2A^*_2+\left[A_1A^*_2e^{i(\mathbf{q}\cdot\mathbf{r}-\delta t)}+\mathrm{c.c.}\right]\right],
\end{equation}
where I have defined the wavevector difference
\begin{equation}
  \mathbf{q}=\mathbf{k_1}-\mathbf{k_2},
\end{equation}
and the frequency difference
\begin{equation}
  \delta=\omega_1-\omega_2.
\end{equation}

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/silberberg_setup.pdf}
  \end{center}
  \caption[Two-beam coupling in a nonlinear medium]{Two-beam coupling in a nonlinear medium. Two waves with frequency $\omega_1$ and $\omega_2$ propagate in a nonlinear medium with angle $\theta$ between their wavevectors $k_1$ and $k_2$.}
  \label{fig:silberberg_setup}
\end{figure}

The first two terms on the right hand side of Eq.~(\ref{eqn:intensity_pattern}) describe a uniform bias of the nonlinear index, and the other term (and its complex conjugate) represent an intensity pattern that is established in the medium by the combination of the two waves. These terms may also be called holographic terms \cite{Silberberg_1984aa} in reference to the storage of wave interference information. As illustrated in Fig.~\ref{fig:intensity_pattern}, the intensity pattern is oriented to bisect $\theta$ and moves in the direction indicated depending on the sign of the detuning $\delta$. In this model, no energy transfer accompanies this scattering if the nonlinear material responds instantaneously to the applied field or if the waves have the same frequency, \emph{i.e.} $\delta=0$. Qualitatively, this can be understood in terms of the relative phase between the intensity pattern and the refractive index grating. If the medium responds instantaneously to the applied field, then the intensity-dependent refractive index grating will exactly follow the intensity pattern and will thus be in phase with the incident waves. For this case, there is equal scattering from one wave, off the grating, and into the other wave. Hence, the nonlinear medium influences only the phase of the interacting fields. In the case of a non-instantaneous nonlinear response and with non-zero frequency difference between the waves, the phase of the index grating differs from that of the intensity pattern and can lead to energy exchange by scattering light from one wave into the other. In this case one wave experiences positive gain while the other experiences negative gain (loss).

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/intensity_pattern.pdf}
  \end{center}
  \caption[Intensity pattern formed by interacting waves]{An intensity pattern forms due to the interference of two waves within a nonlinear medium. The intensity pattern gives rise to a refractive index grating that moves with a phase velocity that depends on $\delta$, the frequency detuning between the waves.}
  \label{fig:intensity_pattern}
\end{figure}

Quantitatively, this can be shown by assuming that the nonlinear part of the refractive index obeys a Debye relaxation equation of the form
\begin{equation}
  \label{eqn:debye}
  \tau\frac{dn_\mathrm{NL}}{dt}+n_\mathrm{NL}=n_2I,
\end{equation}
which has the steady state solution that is familiar from Chapter~\ref{cha:opt_switching}: $n_\mathrm{NL}=n_2I$. % Equation~\ref{eqn:debye} can be solved to give the result
% \begin{equation}
%   \label{eqn:debye_n}
%   n_\mathrm{NL}=\frac{n_2}{\tau}\int_{-\infty}^{t}I)t')e^(t'-t)/\tau)dt'.
% \end{equation}
% Inserting Eq.~\ref{eqn:intensity_grating} into Eq.~\ref{eqn:debye_n} shows that the nonlinear contribution to the refractive index is given by
By considering transient conditions via Eq.~(\ref{eqn:debye}), and requiring that the field in the medium satisfy the wave equation with refractive index
\begin{equation}
  n=n_0+n_\mathrm{NL},
\end{equation}
the spatial variation of the intensities obey
\begin{align}
  \label{eqn:two_beam_intensity}
 \frac{dI_2}{dz}&=\frac{2n_2\omega}{c}\frac{\delta\tau}{1+\delta^2\tau^2}I_1I_2\\
 \frac{dI_1}{dz}&=-\frac{2n_2\omega}{c}\frac{\delta\tau}{1+\delta^2\tau^2}I_1I_2.\\
\end{align}
For the case of a positive value of $n_2$, Eq.~(\ref{eqn:two_beam_intensity}) predicts $I_2$ will experience gain for positive $\delta$, \emph{i.e.} for $\omega_2<\omega_1$. Note that for $\delta=0$, or in the limit of an infinitely fast nonlinearity $\tau\rightarrow0$, the coupling of intensity between the two waves vanishes \cite{Boyd_2002aa,Silberberg_1982aa,Silberberg_1984aa}.

Even for the case of optimal coupling between the two waves, the instability described by this model does not give rise to waves that propagate in new directions. Although the intensities of the waves may undergo oscillations or exhibit chaos \cite{Silberberg_1982aa}, this model alone is not sufficient to explain the generation of off-axis light from beams counterpropagating in a nonlinear medium. In the following section, I describe the treatment given by Yariv and Pepper \cite{Yariv_1977aa} which includes a second pair of counterpropagating beams oriented at an angle to the first pair, and demonstrates off-axis gain arising from four-wave mixing. Allowing gain in an off-axis direction is fundamental to describing optical pattern formation, in fact transverse patterns will result spontaneously if the gain becomes large enough that quantum fluctuations grow in to macroscopic fields.

% section silberberg_and_bar_joseph (end)

\section{Degenerate four-wave mixing} % (fold)
\label{sec:degenerate_four_wave_mixing}

Consider the situation illustrated in Fig.~\ref{fig:yariv_pepper}, where two pump waves counterpropagate through a nonlinear medium along the $z$-axis and a second pair of waves counterpropagate at an angle through the medium. The fields are taken to be degenerate plane waves with frequency $\omega$, and can be represented by
\begin{align}
  \widetilde E_i(\mathbf{r},t)&=E_i(\mathbf{r})e^{i\omega t}+\mathrm{c.c.}\\
  &=A_i(\mathbf{r})e^{i(\mathbf{k_i}\cdot\mathbf{r}-\omega t)}+\mathrm{c.c.},
\end{align}
for $i=1,2,3,4$, where $A_i(\mathbf{r})$ describes the slowly varying envelope for each wave. With counterpropagating waves, we have
\begin{equation}
  \mathbf{k_1}+\mathbf{k_2}=0,\quad \mathbf{k_3}+\mathbf{k_4}=0.
\end{equation}

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/yariv_pepper.pdf}
  \end{center}
  \caption[Geometry of four wave mixing]{Four wave mixing in the phase conjugate geometry. Two strong pump waves with amplitudes $A_1$ and $A_2$ counterpropagate through a nonlinear medium and at an angle to the $z$-axis. An incident wave propagating along $z$ with amplitude $A_3$ undergoes phase conjugate reflection and a fourth wave with amplitude $A_4$ exits the medium propagating along $-z$.}
  \label{fig:yariv_pepper}
\end{figure}

\begin{figure}[htbp]
  \centering
    \includegraphics[scale=1]{Figures/four_wave_mixing.pdf}
  \caption[Four-wave mixing in a nonlinear medium]{Four-wave mixing in a nonlinear medium. a) An intensity grating established by the interference of one pump wave, incident from the right, and the signal wave, incident from the left, creates a refractive index grating. b) The second pump wave, incident from the left, scatters off the index grating to provide the conjugate beam which exits the medium to the left.}
  \label{fig:four_wave_mixing}
\end{figure}

It is important to note that these are degenerate waves, and as such, the treatment of Silberberg and Bar-Joseph predicts that the nonlinear medium induces only a bias phase shift, and does not effect the intensity of the pump waves. Hence it is reasonable to ignore the modification of the pump waves due to the nonlinear interaction. Doing so leads to the following four expressions for the nonlinear polarization produced by the fields in the medium
\begin{equation}
  \label{eqn:polarizations}
  P_1=P_2=0,\quad P_3=6\chi^{(3)}E_1E_2E_4^*,\quad P_4=6\chi^{(3)}E_1E_2E_3^*.
\end{equation}
Each of the interacting waves obeys the wave equation in the form
\begin{equation} 
  \label{eqn:wave_eqn} \nabla^2\widetilde E_i-\frac{n^2}{c^2}\frac{\partial^2\widetilde E_i}{\partial t^2}=\frac{4\pi}{c^2}\frac{\partial^2\widetilde P_i}{\partial t^2},
\end{equation}
where we see how the nonlinear polarization $\widetilde P_i$ acts as a source term driving the wave equation. Physically, this indicates the role of the polarization in generating new waves. In general, the generated waves may have frequencies and wavevectors that are not present in the incident fields. It is clear that each term $P_i$ in the polarization, Eq.~(\ref{eqn:polarizations}), drives the wave equation and gives rise to waves that couple to $E_i$. Thus, the nonlinear polarization of the medium plays the role of coupling the four interacting waves. This coupling can be determined by solving Eq.~(\ref{eqn:wave_eqn}) in the slowly varying amplitude approximation, and neglecting the depletion of the pumps. The result is a pair of amplitude equations for the signal and conjugate waves
\begin{align}
  \frac{dA_3}{dz}&=i\kappa A_4^*\\
  \frac{dA_4}{dz}&=-i\kappa A_3^*,
\end{align}
where
\begin{equation}
  \kappa=\frac{12\pi\omega}{nc}\chi^{(3)}A_1A_2,
\end{equation}
is the coupling coefficient.

A physically interesting case is that of a single input $A_3(0)$ at $z=0$, and $A_4(L)=0$. This corresponds to injecting both pump waves, and the signal wave, in order to observe the output spontaneously generated as the fourth wave, which is typically called the conjugate wave for this geometry. In this case the reflected (conjugate) wave at the input ($z=0$) is
\begin{equation}
  \label{eqn:pco_reflect}
  A_4(0)=i\left(\frac{\kappa}{|\kappa|}\tan |\kappa|L\right)A_3^*(0),
\end{equation}
and the transmitted wave at $z=L$ is
\begin{equation}
  \label{eqn:pco_transmit}
  A_3^*(L)=\frac{A_3^*(0)}{\cos|\kappa|L}.
\end{equation}
The most significant implication of this result is that for the case where $|\kappa|L=\pi/2$, the gain for the transmitted wave, Eq.~(\ref{eqn:pco_transmit}) is infinite as is the conjugate reflectivity, Eq.~(\ref{eqn:pco_reflect}). Physically this implies that the presence of even a single photon in the mode corresponding to $A_3(0)$ is sufficient to generate a macroscopic field at $A_3(L)$. This infinite gain (and reflectivity) is one origin off-axis beams generated by four-wave mixing processes. This analysis has only allowed for four plane waves interacting in the system, yet conditions exist whereby quantum fluctuations are sufficient to induce new beams of light. The final sections of this chapter describe one further step in modeling the counterpropagating beam system: the inclusion of the transverse components of the fields.

% section yariv_and_pepper (end)

\section{Transverse instability} % (fold)
\label{sec:transverse_instability}

The two previous treatments of counterpropagating waves, have explicitly ignored the transverse degrees of freedom by assuming only plane-wave optical fields. In a system similar to those considered above, Firth and Par\'e derive amplitude equations, including transverse components of the field, for counterpropagating fields in a nonlinear medium. Including the transverse degrees of freedom lifts the requirement that the medium respond with characteristic time $\tau$ in order to exhibit instability \cite{Firth_1988aa, Silberberg_1984aa}. Furthermore, with the inclusion of transverse components, the threshold for self-oscillation can be significantly lower than that originally shown by Yariv and Pepper. The model presented by Firth and Par\'e is discussed in the following sections, and serves as the basis for numerical simulations presented in Chapter~\ref{cha:nummodel}.

\subsection{Model of Firth and Par\'e} % (fold)
\label{sub:model_of_firth_and_pare}

The experimental system considered here is familiar from earlier sections, and illustrated in Fig.~\ref{fig:firth_pare}. Two waves counterpropagate through a nonlinear medium and are described by
\begin{align}
  \label{eqn:fields}
  \widetilde E_F\rt&=E_1\rt e^{-i\omega t} +\cc\nonumber\\
  &= F\rt e^{i(kz-\omega t)}+\cc\\
  \widetilde E_B\rt&=E_2\rt e^{-i\omega t} +\cc\nonumber\\
  &= B\rt e^{i(-kz-\omega t)}+\cc,
\end{align}
where $k=n_0\omega/c$, $E_i$ is a field amplitude with spatial dependence $e^{ikz}$, and $F (B)$ is a slowly varying amplitude describing the forward (backward) wave. To study the origin of transverse pattern formation, we consider waves that satisfy Maxwell's equations under the following simplifying assumptions. If the nonlinear medium under consideration is isotropic, dispersionless, and the waves propagate primarily along one axis, the wave equation can again be taken to have the form shown in Eq.~(\ref{eqn:wave_eqn}).
% \begin{equation} 
%   \label{eqn:wave_eqn} \nabla^2\widetilde{E}-\frac{\epsilon^{(1)}}{c^2}\frac{\partial^2\widetilde{E}}{\partial t^2}=\frac{4\pi}{c^2}\frac{\partial^2\widetilde{P}^{\mathrm{NL}}}{\partial t^2},
% \end{equation}
% where $\epsilon^{(1)}$ is a scalar quantity that describes the proportionality between the linear part of the displacement and the electric field $\widetilde D^{(1)}=\epsilon^{(1)}\widetilde{E}$, and $\widetilde{P}^{\mathrm{NL}}$ is the nonlinear part of the polarization vector.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/firth_pare.pdf}
  \end{center}
  \caption[Counterpropagating beams in a nonlinear medium]{Two waves with amplitudes $F(\mathbf{r},t)$ and $B(\mathbf{r},t)$ counterpropagate within a nonlinear medium.}
  \label{fig:firth_pare}
\end{figure}

To go from the wave equation to a quantitative description of the field amplitudes in a counterpropagating beam system, we substitute Eq.~(\ref{eqn:fields}) into Eq.~(\ref{eqn:wave_eqn}) and match terms that exhibit the necessary spatial dependence required to couple to one pump beam or the other. In this sense, the present analysis is very similar to that of Silberberg and Bar-Joseph, in that we study the amplitude equations of two counterpropagating fields. The primary difference is that the medium may now respond instantaneously and transverse variations in the waves are kept in the model.

There are many possible terms that contribute to the polarization $\widetilde P_i$, but most of them do not couple strongly to either of the pump waves and are thus neglected. The two terms that provide coupling between the pump waves are
\begin{align}
  \label{eqn:firth_p_terms}
  P_1&=3\chithree\left(E_1^2E_1+2E_1E_2E_2^*\right)\\
  P_2&=3\chithree\left(E_2^2E_2+2E_2E_1E_1^*\right),
\end{align}
where $P_1$ ($P_2$) generates a wave that couples to $E_1$ ($E_2$). Using the expressions for the incident waves Eq.~(\ref{eqn:fields}), these polarization terms, when substituted into Eq.~(\ref{eqn:wave_eqn}) yield amplitude equations for the forward and backward fields
\begin{align}
  \label{eqn:field_amps}
  \frac{\partial F}{\partial z}+\frac{n_0}{c}\frac{\partial F}{\partial t}-\frac{i}{2k}\frac{\partial^2 F}{\partial x^2}&=i\left[|F|^2+2|B|^2\right]F,\\
  -\frac{\partial B}{\partial z}+\frac{n_0}{c}\frac{\partial B}{\partial t}-\frac{i}{2k}\frac{\partial^2 B}{\partial x^2}&=i\left[2|F|^2+|B|^2\right]B,
\end{align}
where the nonlinear constant $n_2$ has been scaled into the field amplitudes.

% subsection transverse_model (end)

\subsection{Linear Stability Analysis} % (fold)
\label{sub:linear_stability_analysis}

To determine the stability of the field amplitudes Eqs.~(\ref{eqn:field_amps}), we consider perturbations about the steady-state plane-wave solutions $F_0(z)$ and $B_0(z)$ of the form
\begin{equation}
  \label{eqn:perturbation}
  F(x,z,t)=F_0(z)\left[1+\epsilon f_+(x,z)e^{\lambda t} + \epsilon f_-^*(x,z)e^{\lambda^* t} \right],
\end{equation}
and a similar equation for $B(x,z,t)$. The form of these perturbations (\emph{i.e.} the inclusion of the conjugate terms) is necessary as the interaction, Eq.~(\ref{eqn:firth_p_terms}), mixes conjugate terms. The transverse nature of these perturbations can be quantified by Fourier decomposition where
\begin{equation}
  \frac{\partial^2}{\partial x^2}f_\pm\simeq -K^2f_\pm,\quad
  \frac{\partial^2}{\partial x^2}b_\pm\simeq -K^2b_\pm.
\end{equation}
Substituting Eq.~(\ref{eqn:perturbation}) into Eqs.~(\ref{eqn:field_amps}) and linearizing about the steady state solutions gives a set of four coupled equations for the perturbation amplitudes $f_\pm$ and $b_\pm$. Whenever these four equations have a nontrivial solution with $\mathrm{Re}\,\lambda>0$ then instability will occur. The boundary between stable states and unstable states is known as the instability threshold. Above threshold the system is unstable and perturbations grow, while below threshold the system is stable and perturbations die out and the system returns to its steady state. For this instability, the threshold curve is shown in Fig.~\ref{fig:threshold_intensity}. This plot can be understood to indicate the boundary between stable and unstable states of the system. The vertical axis represents increasing intensity or increasing length. The horizontal axis represents the transverse component of the perturbing wavevector. Thus, from this plot, we see that as the intensity is increased (imagine moving a horizontal bar up from the $IL=0$ axis), the first point we reach on the curve indicates that the transverse wavevector with the lowest threshold for self-oscillation corresponds to $K^2L/2k\simeq3$, with $IL\simeq 0.45$. Other wavevectors have higher self-oscillation thresholds, and it is also important to note that the plane-wave threshold ($K=0$) is infinite, in agreement with the results of Silberberg and Bar-Joseph for the case of an instantaneous nonlinear response. Also plotted in Fig.~\ref{fig:threshold_intensity} is the threshold predicted by Yariv and Pepper, $IL=\pi/4$. Clearly the transverse instability occurs at a lower overall threshold, but note that in the limit of large $K$, the threshold curve approaches $IL=\pi/4$.

\begin{figure}[htbp]
  \centering
    \includegraphics[height=3in]{Figures/threshold_intensity.pdf}
  \caption[Threshold intensity for self-focusing media with phase grating]{Threshold intensity for self-focusing media with phase grating, from \cite{Firth_1988aa}. The vertical axis is in units of $IL$, and the horizontal axis is in units of $K^2 L/2k$ where $K$ is the wavevector of the perturbation.}
  \label{fig:threshold_intensity}
\end{figure}

Finally, it is instructive to apply these results to a physical situation. As an example, consider a rubidium system. The prediction that self-oscillation occurs at $K^2L/2k\simeq3$ implies that for a 5~cm sample of rubidium vapor, $\lambda=$780~nm, the angle between the waves with wavevector $K$ and the pump waves is approximately $\theta\simeq K/k\simeq3$~mrad. In the next chapter, I present a pattern forming system consisting of two pump waves that counterpropagate in rubidium vapor. It should not be surprising that this system generates a cone of light that propagates at an angle to the pump waves of $\simeq3$~mrad.

% subsection linear_stability_analysis (end)

% section firth_and_par'e (end)

% chapter polarization_instability_in_rb_vapor (end)